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That graph says a lot more than most probably realise:
1. The rise in CO2 concentration is almost linear since the early 1700's despite very minimal amounts of fossil fuels being used at the time. Coal use was just beginning then with oil not coming until the late 1850's and significant use of gas not until the 1930's.
And more to the point, total fossil fuel use remained low at the global level until after WWII, a situation that makes a linear rise in CO2 concentration seem somewhat odd.
Whilst it seems logical that burning fossil fuels would increase CO2 in the atmosphere, that graph shows that most of the rise in CO2 happened before most of the fossil fuel burning occurred. Regardless of the climate change issue, that trend of itself does warrant some proper scientific (not political) investigation to find out what is going on. It clearly seems that the rise in CO2 is not due to fossil fuel use, and certainly not due to oil or gas which were not in use until after half the rise had already happened.
2. The graph shows a relationship between rising CO2 and rising temperature existed during the period 1700 to 1800 and again from about 1900 to 1930, and again during sometime around 1980 to the late 1990's.
So, from 1200 years of data we find a relationship between rising CO2 concentration and rising temperature occurs for about 150 years or 12.5% of the time. For the other 1000+ years and 87.5% of the time, CO2 and temperature do not change in parallel and often head in different directions at the same time, a situation which suggests that CO2 is not the dominant determinant of the earth's temperature. Whether or not it has any effect at all can not be deduced from that graph.
But what about the spike during the late 20th Century that was more rapid than previous temperature changes? That is most likely a function of more accurate measurements.
An analogy to illustrate the point. Suppose that you live in Sydney (or anywhere else, I'll just pick Sydney for example) and you have a thermometer at home.
Using your thermometer, you can get very accurate data on temperature changes as often as you want. Plot these on a graph and you'll get a chart that's all over the place with highs and lows.
Now go back and add historic data since European settlement in 1788. Looking through the record books, you'll have real trouble finding anything more than daily data and to a large extent you will end up settling for monthly or annual averages. Now your chart will look a lot smoother - there probably were heatwaves in 1800 but your data doesn't show this, hence your chart won't show it either.
Now add some data for the previous 1000 years based on some proper research. No way is that going to show daily or even annual temperature. At best, you'll be able to approximate general conditions over a decade or more and base your chart on that. Now your chart has no spikes at all because you have no evidence that they ever occurred, although it would be fair to assume that it wasn't 22 degrees every single day 1000 years ago.
So in short, the chart shows abrupt changes recently not because the climate has suddenly started abruptly changing, but because we now have the means to accurately measure it.
If I plot the price of BHP shares over the past 10 years, and then go back and get annual end of year data since the company was formed, then that too will show a recent spike in voltatility simply because I haven't included daily or monthly data until quite recently. Same with anything.
Overall, that graph is pretty convincing that CO2 isn't something to worry too much about in terms of the climate. But we ought to investigate the relationship between CO2 concentration and fossil fuel use to see why it does not behave as one would logically expect.
The rate of change of CO2 concentration does not parallel the rise in CO2 emissions as one would logically expect. That raises a lot of questions (1) why and (2) would reducing CO2 emissions by 20% or 50% (for example) have any effect on CO2 concentrations given that CO2 concentrations have risen at much the same rate despite huge changes in emission levels over the 300 years that CO2 concentraton has been rising at a roughly linear rate.
That situation suggests there's more to the story than a simplistic relationship between man-made CO2 emissions and CO2 concentration in the atmosphere - we emit more and more but has shown up in atmospheric measurements once we passed the relatively trivial emission levels of the 1700's.
In that case any cut to fossil fuel use, unless it is 100% or very close to it, would seem to be pointless even if there were a reason to be worried about CO2 in the atmosphere.